Messing Around with Math

February 3, 2016

Here’s a puzzle that my office mate shared over the cubicle wall last week.


Now, Guttman is unconventional in the sense that I share cubicle space with English professors, a librarian, secretaries to the Deans, an academic advisor, a sociologist, and so on. Two of these “non-math” people immediately became obsessed with this problem. They spent every free moment of the next two days drawing stars and lines until they’d found a workable solution.

This is basically a math problem. Let’s a consider a second, very different math problem and the very different reactions it might evoke.

The profit for a certain brand of MP3 player can be described by the function P(x) = 40x – 3000 – 0.01x^2 dollars, where x is the number of MP3 players sold. To maximize profit, how many MP3 players must be produced and sold?

I took this problem from what I consider to be a top-notch College Algebra textbook, but it’s pretty difficult to imagine my office mates rushing to find the optimal numbers of MP3 players. What makes these two problems and the reactions they evoke so different? How do we get our students engaged in the second type of problem?

It might be worth stepping back for a moment to consider the problem-solving strategies our students employ. I’ve been reading this book.


Through three years of collaborative, ethnographic work, Mizuko Ito and fourteen other researchers have identified three major “genres of participation”: “hanging out,” “messing around,” and “geeking out.” These genres describe how young people use digital technology. It’s important to note that these are genres of participation, not genres of people. Instead of casting individuals into broad categories like “geek” or “luddite,” they acknowledge that one person can participate as a geek in one setting and a luddite in another. Our identities are neither singular nor fixed.

The genre of “hanging out” refers to friendship-driven usage of digital technology, activities like sharing photos from a party on Facebook or exchanging texts with a friend. On the other hand, “messing around” represents the first steps into more interest-driven uses of technology. This category includes what Ito et al. call “fortuitous searching,” “moving from link to link, looking around for what many teenagers describe as ‘random’ information” (pg. 54). Note how this type of search is open-ended rather than goal-driven. In general, “messing around” describes casual, open-ended experimentation with new media that’s guided by interests in a certain topic, but does not necessarily have a specific end-goal. Finally, “geeking out” refers to intense, interest-driven engagement with new media such as maintaining a blog about a particular interest or editing a video. It tends to be less open-ended than “messing around” and requires a more deliberate investment on the part pf the participant.

When my office-mates engrossed themselves in the stars and triangles problem above, they were “messing around” with math. Although it ended up consuming a lot of their time, the puzzle didn’t require any sort of “geeky” investment. There is something about this puzzle that lends itself to “fortuitous searching,” experimenting with different arrangements of lines without knowing exactly what you’re after. Compare this with the MP3 player problem, which has a very clear goal (the optimal number of MP3 players to produce) and cannot be solved by simply playing with lines on a page. You have to “geek out” on this problem or there’s no hope.

Goal-oriented tasks are difficult for my students. They don’t always have the metacognitive skills to visualize the end-goal. Even if a student has a casual interest in mathematics, it can be difficult to keep them focused enough to persist and solve a difficult problem. They may start the MP3 player problem, realize that it requires them to coordinate a lot of different ideas, and then they lose interest.

I wonder if “messing around” could be the gateway drug to “geeking out” about math. How would you do this? Let’s say I have a list of major learning objectives for my math class, and one of them is “identifying the vertex of a parabola.” I need students to master this specific concept so I cannot have them just “messing around” with any idea that piques their interest. Is there a “messing around” activity that I can create, which will help students achieve this learning objective?

It seems like a good first step would be identifying precisely what differentiates these two types of tasks. Then, I can take something like the MP3 player problem and craft it’s features to be more like the stars and triangles problem. When I was talking about this with my office-mate, he mentioned casual games like Threes or Theseus and the Minotaur.


What makes these games so addictive? How is that people can swallow hours and hours of time in them and yet it feels like no big investment to pick them up and play them? Here’s what I’ve come up with so far; I’d love to hear your ideas as well.

  1. The “messing around” tasks require very little domain-specific knowledge. You don’t need to know what a quadratic function is or a vertex to understand the stars and triangles problem. You don’t need to have any fluency with algebraic symbols and manipulations; the meaning of the problem is immediately clear and so are the tools with which you’re working.
  2. Each time you fail a “messing around” task, you always feel like you were just about to get it. You draw some lines on the star, count up the triangles, and realize you’re two short of ten, but you’re almost there! Next time, you’ll get it. This motivates you to keep on trying; the answer is just around the corner! Every time I play Threes, I’m convinced that this will be the time I get the 1024 block. On the other hand, when you get stuck on the MP3 problem, there’s the sense that you haven’t made any progress and you don’t know what you’re doing. You feel lost. You never feel lost on a “messing around” task.
  3. A “messing around” task allows for “fortuitous searches.” You can stumble around and experiment without knowing exactly where you’re going or what you’re after. I never have a very deliberate strategy in mind when I play Threes, but that quadratic function problem, you need to stay very focused on the goal. I think this is closely related to (or equivalent to) the previous property because by definition, fortuitous searches always give you the feeling that you’re just about to find what you’re looking for.
  4. They’re portable. This is really important. There’s a reason why mobile phone games tend to be more casual and puzzle-like. If participation requires you to be in a specific place at a specific time then it requires another level of determination to engage with the task. The beauty of the stars and triangles problem is that you can work on it anywhere, doodling on a napkin or a piece of paper while you’re chatting with a friend. Speaking of which…
  5. A “messing around” task does not require your full attention. You can play Threes while listening to music, while riding on the subway, while watching TV, while talking to a friend; it can ride on autopilot in the background while you do other things. There is no autopilot for the quadratic functions problem above. This could be an important property as Ito et al. note that “certain forms of participation also act to bridge the divide between friendship-driven and interest-driven modes. In chapter 5, we will describe how friendship-driven modes of ‘hanging out’ with friends while gaming can transition to more interest-driven genres of what we call recreational gaming” (pg. 17). Maybe, “hanging out” with friends while solving math puzzles can transition to more serious mathematical problem-solving.

There must be other properties that I have not thought of yet, or maybe some of these properties are not as essential as they seem? I worry that property 2 means students will keep trying the same false strategy (because they’re just about to get it) and thus they’ll persistent with bad strategies rather than reflecting on and finding new ones.

Nonetheless, my idea is that you might take a goal-oriented problem like the MP3 player example and modify it according to the five properties listed above, creating something more appropriate for “messing around.” Once a student’s interest is piqued, you start slowly shedding these five properties. Maybe it becomes less portable as you introduce more complicated elements. Maybe, you introduce domain-specific terminology and tools like variables and functions, or maybe you give them more and more specific goals to work towards.What do you think?